Kindly solve it urgently

47. 49. 50. 51. 52. 53. 54. 55. 56. ban-.111 . Suppose that g and h induce the same inner automorphism of a group G. Prove that h'lg E 2(6). . Combine the results of Exercises 45 and 47 into a single \"if and only if\" theorem. If as and [3 are elements in .5"1 (n a 3),prove that (pa : (,bB implies that o: : B. (Here. qba is the inner automorphism of Sn induces by as.) Prove or disprove that the mapping (,6 from Q+, the positive rational numbers under multiplication, to itself given by d)(.::) : x2 is an automorphism. Suppose the d) and 'y are isomorphisms of some group G to the same group. Prove that H 2 {g E G | (Mg) 2 7(3)} is a subgroup of G. Let G be a group. Complete the following statement: |Inn(G)| : I if and only if Suppose that G is an Abelian group and d; is an automorphism of G. Prove that H 2 {x E G I 950:) = x'l} is a subgroup of G. Let (1') be an automorphism of 133' What are the possibilities for $(R45)? Let d) be an automorphism of C\